Notions of numerical Iitaka dimension do not coincide

Abstract

Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $\mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $\mathbb R$-divisor $D_+$ for which $h^0(X,\lfloor m D_+ \rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.